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Valley Degeneracy in Biaxially Strained Aluminum Arsenide Quantum Wells

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Valley Degeneracy in Biaxially Strained Aluminum Arsenide Quantum Wells PHYSICAL REVIEW B 84, 125319 (2011) Valley degeneracy in biaxially strained aluminum arsenide quantum wells S. Prabhu-Gaunkar,1 S. Birner,2 S. Dasgupta,2 C. Knaak,2 and M. Grayson1 1Electrical Engineering and Computer Science, Northwestern University, Evanston, Illinois 60208, USA 2Walter Schottky Institut, Technische Universitat¨ Munchen,¨ Garching DE-85748, Germany (Received 13 September 2010; revised manuscript received 19 May 2011; published 16 September 2011) This paper describes a complete analytical formalism for calculating electron subband energy and degeneracy in strained multivalley quantum wells grown along any orientation with explicit results for AlAs quantum wells (QWs). In analogy to the spin index, the valley degree of freedom is justified as a pseudospin index due to the vanishing intervalley exchange integral. A standardized coordinate transformation matrix is defined to transform between the conventional-cubic-cell basis and the QW transport basis whereby effective mass tensors, valley vectors, strain matrices, anisotropic strain ratios, piezoelectric fields, and scattering vectors are all defined in their respective bases. The specific cases of (001)-, (110)-, and (111)-oriented aluminum arsenide (AlAs) QWs are examined, as is the unconventional (411) facet, which is of particular importance in AlAs literature. Calculations of electron confinement and strain for the (001), (110), and (411) facets determine the critical well width for crossover from double- to single-valley degeneracy in each system. The biaxial Poisson ratio is calculated for the high-symmetry lower Miller index (001)-, (110)-, and (111)-oriented QWs. An additional shear-strain component arises in the higher Miller index (411)-oriented QWs and we define and solve for a shear-to-biaxial strain ratio. The notation is generalized to address non-Miller-indexed planes so that miscut substrates can also be treated, and the treatment can be adapted to other multivalley biaxially strained systems. To help classify anisotropic intervalley scattering, a valley scattering primitive unit cell is defined in momentum space, which allows one to distinguish purely in-plane momentum scattering events from those that require an out-of-plane momentum component. DOI: 10.1103/PhysRevB.84.125319 PACS number(s): 73.21.Fg, 73.50.Bk I. INTRODUCTION Various formalisms have been developed to understand valley occupancy in a multivalley system. AlAs QWs are Calculating valley degeneracy in a quantum well (QW) slightly strained with respect to the GaAs substrate, so one requires a comprehensive treatment of strain, quantum confine- must consider quantum confinement, strain, and piezoelectric ment, and piezoelectric fields since all contribute at compara- fields to estimate the electron subband energy and degeneracy ble energy scales. Of the various multivalley semiconductors, the indirect-bandgap zinc blende semiconductor aluminum of the different electron valleys. Stern and Howard modeled arsenide (AlAs) with its bulk threefold valley degeneracy is two-dimensional (2D) confinement of electron valleys in Si for arbitrary crystal plane orientations neglecting strain of particular interest (Fig. 1) because its heavy anisotropic 21 electron mass allows for large interaction effects,1 and its effects. Van de Walle theoretically studied the absolute energy level for an unstrained semiconductor heterojunction, near-perfect lattice match to GaAs substrates allows for 22 high-mobility, modulation-doped quantum wells (QWs).2,3 considering strain effects only for bulk systems. Smith AlAs/AlGaAs QWs can reach mobilities of the order of μ = et al. calculated the strain tensor theoretically for the case of 23 100 000 cm2/Vs in (001)-facet QWs4 and in the high-mobility (001)- and (111)-oriented QW superlattices. Although Caridi direction of anisotropic (110)-facet QWs.5 Unconventional and Stark derived the complete strain tensor for arbitrarily facets such as (411) have also proven useful in identifying oriented substrates with cubic symmetry, they ignored a exchange effects like quantum Hall ferromagnetism in AlAs critical shear component, which arises in the high Miller 24 QWs.6,7 Evidence for a QW width crossover from double- index directions under relaxation. De Caro et al. calculated to single-valley occupation has been shown for (001)AlAs the shear strain with commensurability constraint equations wells,8,9 as has evidence for single-valley occupancy in wide accurately only for the low index (100), (110), and (111) (110)AlAs wells.5 Dynamic control of the valley degeneracy has been realized with uniaxially strained (001)AlAs QWs to induce valley degeneracy splitting.1,10,11 Such studies can quantify interaction effects, calibrating valley strain suscepti- bility and valley effective mass.10,12–14 Quantum confinement to a one-dimensional multivalley system has been achieved in cleaved-edge overgrown quantum wires15,16 and quantum point contacts.17 Novel interaction effects in QWs include anisotropic composite Fermion mass18 and valley skyrmions, whereby electrons populate linear superpositions of two FIG. 1. (Color online) (Left) Real-space depiction of AlAs QW 19 valleys at once. Such interaction effects that result from structures defining the transport basis axes: a,b, and c. (Right) 20 exchange splitting of a perfect SU(2) symmetry may prove Brillouin zone for bulk AlAs defining the conventional-cubic-cell useful in future quantum device applications, where the valley basis x,y, and z. Note the three degenerate valleys that are occupied degree of freedom functions as a pseudospin. with electrons as indicated by ellipsoidal equienergy contours. 1098-0121/2011/84(12)/125319(15)125319-1 ©2011 American Physical Society S. PRABHU-GAUNKAR et al. PHYSICAL REVIEW B 84, 125319 (2011) orientations.25 Yang et al. corrected the constraint equations strain tensor, degeneracies, piezoelectric fields, and valley- and calculated piezoelectric fields arising from the shear splitting energies for the specific cases of (001)-, (110)-, strain components for the general case of a pseudomorphic (111)- as well as the unconventional (411)-oriented QWs film.26 Adachi calculated the strain tensor theoretically for illustrate how valley degeneracies can be engineered. We the case of bulk and superlattice structures but did not review the crossover width calculation for double-to-single address the case of single QW structures.27 Hammerschmidt valley occupation in the (001)-, (110)-, and (411)-facet QWs. et al. focus on the case of isotropic planar strain arising in For the high-symmetry low-Miller-index orientations (001), QWs but do not carry out a combined treatment of strain (110), and (111), we calculate the biaxial Poisson ratio, and for energy along with quantum confinement energy and how these the low-symmetry high-Miller-index (411) facet, there arises affect valley degeneracies.28 Theoretically, Rasolt proposed a shear component in the strain tensor and, consequently, a an understanding of the various continuous symmetries and shear-to-biaxial strain ratio. We calculate intervalley scattering symmetry breaking in a multivalley system in terms of an ratios for valley-degenerate AlAs QWs on various facets SU(N) symmetry, where N is the valley degeneracy number,20 in Sec. VIII and refer to the valley-scattering unit cell for but did not explicitly derive the vanishing intervalley exchange intuition. We complete our analysis by comparing the standard integral, which underlies this theory. Material-specific calcu- 2D Bravais lattice valley representation to our valley cell lations of quantum confinement and strain in high-mobility representation. To generalize the formalism for calculating valley subband energies in arbitrarily oriented facets, we end Si and Ge structures are well studied for high-symmetry 43 facets29–40 and, recently, such calculations have been been in Sec. IX by adapting this notation for miscut samples. made in other quantum confined systems as well.41,42 However, a combined treatment of quantum confinement, strain, and II. VALLEY EXCHANGE INTEGRAL AND VALLEY INDEX piezoelectric fields to determine valley degeneracy in QWs is AS A PSEUDOSPIN lacking, especially, for low-symmetry facets. Just as electrons with different spins are distinguishable and To eventually model transport in such valley-degenerate have no exchange term in their interaction energy, electrons in systems, one must also consider the extra scattering channel different valleys can be shown to be effectively distinguishable not present in single-valley systems, namely, intervalley from each other due to the vanishingly small intervalley scattering. It is therefore useful to introduce a k-space unit cell exchange energy. The valley index can thus be treated as that permits visualization of momentum-scattering events in a a pseudospin index. Rasolt20 discusses symmetry breaking geometry that is natural to the quantum-confinement direction. in a multivalley system but does not explicitly derive this However, the standard depiction of a 2D Brillouin zone of exchange integral. Previous work on Si quantum dots to study a quantum-confined valley-degenerate system21 projects all Kondo effect44–46 calculates the exchange integral only for the valleys to a single 2D plane. Such a depiction loses information special case of identical spatial wavefunctions on Si quantum about the out-of-plane momentum-scattering component that dots. For completeness, in this section we explicitly derive was projected out, which is necessary to determine the
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